Notebooks and Worked Examples

These notebooks turn the formulations of quantum dynamics into reproducible calculations. Each article specifies a model, conventions, algorithm, analytic or independently computed benchmark, convergence study, failure modes, and validation targets. The aim is not merely to produce plausible plots. It is to establish why a computed result should be trusted.

The articles are implementation guides rather than canonical homes for the underlying physics. Formal definitions and derivations remain in their subject chapters. A notebook may restate the minimum formula needed to implement a test, but it links back to the page that owns the theorem, propagator, phase-space construction, or approximation.

What This Chapter Owns

This chapter is the canonical home for reproducible numerical workflows that accompany quantum dynamics:

• Fourier-grid evolution and wave-packet diagnostics;
• operator, state, and observable comparisons among quantum pictures;
• kernel convolution and spectral reconstruction of propagators;
• ordered products, Trotter formulas, and time-stepping convergence;
• normalized imaginary-time projection and symmetry-sector traps;
• discrete Wigner transforms, marginal checks, purity, and negativity;
• one-period Floquet operators, quasienergy branches, and stroboscopic validation;
• shared standards for baselines, error metrics, refinement, and reproducibility.

The chapter does not own general numerical analysis, software installation, or the formal theory being simulated. Floating-point behavior, matrix algorithms, ODE methods, FFT conventions, and convergence methodology belong to the Mathematical Toolkit. Cross-site notebook metadata and validation policy belong to Software, Notebooks, and Benchmarks.

Shared Computational Contract

A mature notebook should separate five layers:

$$\begin{gathered} \text{physical model and conventions}, \\ \text{finite numerical representation}, \\ \text{algorithm and control parameters}, \\ \text{independent benchmark}, \\ \text{diagnostics and refinement evidence}. \end{gathered}$$

Agreement with intuition is not a validation layer. Neither is conservation of one invariant by itself. A unitary but incorrectly ordered product preserves norm; a periodic FFT calculation can preserve norm while suffering wraparound; a normalized imaginary-time iteration can converge to the wrong symmetry sector.

When an exact result is available, report an error such as

$$e_N = \left\| Q_N-Q_{\rm ex} \right\|,$$

for a refinement parameter $N$. When no exact result is available, compare independent methods or successive refinements and state that the result is a convergence estimate rather than an exact error. For a method expected to have order $p$, test whether

$$\frac{e_N}{e_{2N}} \longrightarrow 2^p.$$

Use operator-level checks when the claim concerns an operator. Testing one state is insufficient because the error may act mainly on an orthogonal subspace.

Page Map

Computational question	Notebook	Primary benchmark
How does a localized free packet translate and spread on a Fourier grid?	Wave-Packet Time Evolution	exact free Gaussian evolution
Do Schrödinger, Heisenberg, and interaction pictures give the same observable history?	Spin Precession in Three Pictures	analytic spin rotation and picture translation
How do full-line kernel convolution and periodic Fourier evolution represent the same free dynamics?	Free-Particle Propagator	exact Gaussian and free kernel
How do a spectral sum, closed kernel, and coherent-state trajectory agree for the oscillator?	Harmonic-Oscillator Propagator	Mehler kernel and coherent evolution
Why does exponentiating an integrated noncommuting Hamiltonian fail?	Time-Dependent Hamiltonian	exact rotating-frame propagator
How do first- and second-order product formulas reveal their convergence orders?	Trotter Evolution	exact finite-dimensional exponential
How does nonunitary imaginary-time evolution isolate low-energy states?	Imaginary-Time Projection	finite-difference diagonalization and oscillator spectrum
How is a continuous Wigner function represented without losing normalization, marginals, or negativity?	Wigner Function	analytic Gaussian and odd-cat parity value
How are eigenphases converted into tracked quasienergy branches and sampled dynamics?	Floquet Quasienergy	exact circular-drive Floquet operator

These nine pages form the planned chapter. Their baselines are deliberately small enough to audit, yet each exposes a failure that also matters in larger research calculations.

Suggested Routes

First numerical dynamics route

Begin with Wave-Packet Time Evolution to connect a continuous wavefunction, a finite Fourier grid, and an exact benchmark. Continue to Spin Precession in Three Pictures for a finite-dimensional operator check. Then use Trotter Evolution to learn refinement and observed-order diagnostics before tackling a genuinely time-dependent Hamiltonian.

Propagator and path-integral route

Read Free-Particle Propagator and Harmonic-Oscillator Propagator alongside the canonical Propagator Kernel and Path Integral Formulation. The free calculation isolates normalization and boundary effects. The oscillator adds spectral truncation, coherent motion, and caustic phases.

Time-stepping and effective-evolution route

Study Trotter Evolution before Time-Dependent Hamiltonian. The first separates algebraic product-formula error from floating-point error. The second makes time ordering unavoidable and compares exact short-step products with an adaptive ODE route. Finish with Floquet Quasienergy to turn a converged one-period propagator into a branch-aware spectrum and long-time discrete map.

Imaginary-time and phase-space route

Use Imaginary-Time Projection to contrast contractive spectral filtering with unitary real-time propagation. Then read Wigner Function to test a representation whose values need not be positive even though its marginals and normalization are physical. The oscillator notebook connects the two routes through coherent-state phase-space motion.

Validation by Failure Mode

Choose a notebook according to the numerical risk, not only the physical model.

Risk to diagnose	Strongest starting page	Decisive check
periodic-boundary wraparound	Wave-Packet Time Evolution	increase box size at fixed resolution
incorrect FFT normalization	Free-Particle Propagator or Wigner Function	integral and marginal identities
phase or picture inconsistency	Spin Precession in Three Pictures	compare the same observable in all pictures
spectral truncation	Harmonic-Oscillator Propagator	increase basis cutoff with a regulator audit
wrong chronological order	Time-Dependent Hamiltonian	compare with the exact rotating-frame operator
claimed convergence order	Trotter Evolution	refinement slope on an operator norm
convergence to a symmetry-restricted state	Imaginary-Time Projection	overlap and parity diagnostics
false positivity assumption	Wigner Function	odd-state origin value and negative volume
quasienergy branch jump	Floquet Quasienergy	unit-circle gap and overlap transport
small per-step phase error at long times	Time-Dependent Hamiltonian or Floquet Quasienergy	repeat evolution after refining one step or period

Common Numerical Language

The pages use a consistent distinction among several errors:

• representation error: finite box, grid, basis, or spectral cutoff;
• discretization error: finite time step, quadrature spacing, or transform grid;
• algorithmic error: splitting, iteration, solver tolerance, or truncation of a formal expansion;
• roundoff error: finite-precision accumulation and conditioning;
• model error: replacing the intended Hamiltonian or domain by a simplified one.

Refining only one parameter can move error from one category to another. Increasing a Fourier grid size at fixed box length improves spatial resolution but does not delay periodic wraparound. Tightening an ODE tolerance does not help if the Hamiltonian is evaluated with the wrong absolute drive phase. Adding oscillator basis states without regulating a real-time spectral sum may make pointwise behavior less stable near a caustic.

Every convergence claim should therefore name what is held fixed and what is refined.

Reproducibility Record

Every completed run should preserve enough information to reproduce the numerical claim:

1. equations, units, basis order, Fourier convention, and boundary conditions;
2. all physical parameters and initial-state definitions;
3. grid endpoints, whether endpoints are included, and array ordering;
4. time-step rule, product order, solver tolerances, and iteration stopping criteria;
5. random seeds and generator family if stochastic sampling is introduced;
6. software language, package versions, and precision;
7. raw scalar diagnostics behind convergence and validation plots;
8. exact or independent reference values with their provenance;
9. warnings, failed checks, and excluded parameter points;
10. the date of the run and a stable identifier for the code revision.

A plot is an output, not a reproducibility record. Preserve the parameters and compact tabular data needed to regenerate it.

Canonical-Home Boundaries

Use these notebooks to implement and test, then return to the canonical concept pages for interpretation:

• wavefunctions, states, and probabilities belong to Core Formalism;
• exact kernels and their composition belong to Propagators and Kernels;
• Schrödinger, Heisenberg, and interaction pictures belong to Pictures of Quantum Dynamics;
• Trotter formulas and quantum maps belong to Operator Dynamics;
• imaginary time and Wigner evolution belong to their formulation chapters;
• Floquet theorems, zones, and stroboscopic interpretation belong to Periodic and Driven Dynamics;
• general software and validation conventions belong to the Reference.

Do not copy a formal derivation into a notebook when a canonical page already owns it. State the formula needed for the computation, declare the convention, and link to the derivation.

Minimum Completion Checklist

Before treating a notebook result as mature, verify that it includes:

• a declared physical and numerical problem;
• at least one analytic or independent benchmark;
• a refinement sequence rather than one favored resolution;
• quantitative error and structural diagnostics;
• an adversarial control that should fail or simplify predictably;
• discussion of boundary, phase, branch, and normalization conventions;
• common failure modes tied to observable symptoms;
• enough environment and parameter information to reproduce the run;
• links to the canonical theory and numerical-method pages;
• references appropriate to the algorithm and physical model.

References

• C. Leforestier et al., “A Comparison of Different Propagation Schemes for the Time Dependent Schrödinger Equation,” Journal of Computational Physics 94, 59–80 (1991), doi:10.1016/0021-9991(91)90137-A.
• H. Tal-Ezer and R. Kosloff, “An Accurate and Efficient Scheme for Propagating the Time Dependent Schrödinger Equation,” Journal of Chemical Physics 81, 3967–3971 (1984), doi:10.1063/1.448136.
• E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration, 2nd ed. (Springer, 2006), doi:10.1007/3-540-30666-8.
• N. J. Higham, Functions of Matrices: Theory and Computation (SIAM, 2008), doi:10.1137/1.9780898717778.

Model-specific and formulation-specific sources are listed on the individual notebook pages.